Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

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Galois group of $x^6 + 3$ isomorphic to a copy of $S_3$ inside $S_6$

I have seen the the thread here related to the computation of the Galois group of the same polynomial. However, my question is not about the computation itself but about the group presentation of the ...
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2answers
401 views

A tower of normal extensions is normal

I am reading Kaplansky's notes on Galois theory. He defines a normal extension as follows: Let $M$ be any field and $K$ any subfield. $M$ is normal over $K$ if for any $u\in M$ but not in $K$, there ...
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Prove that the intersection of all subfields of the reals is the rationals

I'm reading through Abstract Algebra by Hungerford and he makes the remark that the intersection of all subfields of the real numbers is the rational numbers. Despite considerable deliberation, I'm ...
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1answer
112 views

Why is the following map well defined?

Let $H\leq G=\operatorname{Gal}(K/F)$ ($K/F$ is a finite galois extension), why is the following map well defined: $\varphi:G/H\to\Gamma_F(K^H,K)$ defined by $\sigma H\mapsto\sigma|_{K^H}$ ,where ...
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$p$-Sylow subgroups of a group of order $5^3\cdot 29^2$

I need to calculate the $p$-Sylow subgroups of a Galois group with order $5^3 \cdot 29^2$, i.e. $|\mathrm{Gal}(K/F)|=5^3 \cdot 29^2$. I've already established that there is only one 29-Sylow-subgroup ...
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What is Gal($\mathbb{F}_{q^k}/\mathbb{F}_q)$?

I know that if $q=p$ (where $p$ is prime) then Gal($\mathbb{F}_{p^k}/\mathbb{F}_p)$ is cyclic of order $k$. I heard that in general (for $q=p^m$) the galois group is cyclic of the order of the ...
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$K/E$, $E/F$ are separable field extensions $\implies$ $K/F $ is separable

I wish to prove that if $K/E$, $E/F$ are algebraic separable field extensions then $K/F$ is separable. I tried taking $a\in K$ and said that if $a\in E$ it is clear, otherwise I looked at the minimal ...
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1answer
133 views

Multiplicative formula for order of automorphism group

I am reading a proof of the following proposition from Dummit and Foote: Let $E$ be the splitting field over $F$ of the polynomial $f(x) \in F[x]$. Then $$|\textrm{Aut}(E/F)|\leq [E:F]$$ ...
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Calculating the numer of irreducible polynomials of degree $12 $ over $\mathbb{F}_p$

I am trying to do an exercise that asks me to calculate the numer of irreducible polynomials of degree $12 $ over $\mathbb{F}_p$. The exercise have 3 parts and I have done the first two parts that ...
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1answer
523 views

Intermediate fields of cyclotomic splitting fields and the polynomials they split

Consider the splitting field $K$ over $\mathbb Q$ of the cyclotomic polynomial $f(x)=1+x+x^2 +x^3 +x^4 +x^5 +x^6$. Find the lattice of subfields of K and for each subfield $F$ find polynomial $g(x) ...
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511 views

Galois group of a degree 5 irreducible polynomial with two complex roots.

Let $f(x)\in \mathbb{Q}[x]$ be an irreducible polynomial of degree five with exactly three real roots and let $K$ be the splitting field of $f$. Prove that Gal$(K/Q) \cong S_5$. My attempt: The ...
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383 views

Galois Groups of Finite Extensions of Fixed Fields

I am trying to prove the following proposition: Let $L$ be an algebraically closed field, $g \in Aut(L)$ and $K=\{x \in L \; | \; g(x)=x\}$. Show that every finite extensions $E/K$ is a cyclic ...
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What is the prerequisite knowledge for learning Galois theory?

What is the prerequisite knowledge for learning Galois theory? I don't know what a ring is.
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161 views

Finding a Galois extension of $\Bbb{Q}(i)$ isomorphic to $D_4$

This problem has been bothering me for a few days, now. This is not homework, just something I do for my own entertainment. I want to find a Galois extension of $\Bbb{Q}(i)$ which has Galois group ...
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1answer
476 views

Formal derivatives over finite fields.

I am slightly confused about what formal derivatives over finite fields mean. Example 1: Consider $f(x)=x^3-2\in \mathbb{F}_7[x]$. By checking each element of $\mathbb{F}_7$ we easily see that this ...
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541 views

Help with finding the fixed field of a subgroup of the galois group

So basically the question is: Let $K$ be the splitting field of $t^5-2$ (over the rationals). Let $\theta=\sqrt[5]{2}$, and $\eta$ a primitive $5$-th root of unity. First we note that the splitting ...
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How to find irreducible polynomials over $\mathbb{Q}(i)$ with prescribed Galois group?

Here is my recent homework question: For each of the following five fields $F$ and five groups $G$, find an irreducible polynomial in $F[x]$ whose Galois group is isomorphic to $G$. If no example ...
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292 views

Generating Elements of Galois Group

I am trying to prove the following: Let $K=\mathbb{Q}(\sqrt{p_1},\ldots,\sqrt{p_n})$. Show that $K/\mathbb{Q}$ is Galois with Galois group $(\mathbb{Z}/2\mathbb{Z})^n$. I have attached my proof ...
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Linear independence of roots over Q

Let $p_1,\ldots,p_k$ be $k$ distinct primes (in $\mathbb{N}$) and $n>1$. Is it true that $[\mathbb{Q}(\sqrt[n]{p_1},\ldots,\sqrt[n]{p_k}):\mathbb{Q}]=n^k$? (all the roots are in $\mathbb{R}^+$) ...
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For $K$ the splitting field of $x^8+1$ over $\mathbb{Q}$, determine $Gal(K/\mathbb{Q})$.

Let $f(x) = x^8+1$. To determine the Galois group $G$, we first need the splitting field and before that we need to find the zeroes of $f$. So, $\left(re^{i\theta}\right)^8 = 0$ implies $r=1, ...
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Cyclotomic field in proof of quadratic reciprocity

I found a proof of quadratic reciprocity in wikipedia, which I don't quite understand. The link is http://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity. On the last line of the Cyclotomic ...
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246 views

Field extensions, inverse limits, notation and roots of unity

I'm hoping I can get some assistance with a revision problem and also a notational issue I'm not sure about (although it may not be standard). I seem to remember going over this or something similar ...
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319 views

How to describe when a simple extension $F(\alpha)/F$ is Galois in terms of the minimal polynomial of $\alpha$?

I have a question concerning definition in terms of minimal polynomial i.e. if we let $E = F(\alpha)$ be a field extension of $F$ of degree two then how do I describe, in terms of the minimal ...
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Towards the solution of the Problem : Field Extension problem beyond $\mathbb C$ (Question 1)

I am posting this problem in order to break the problem in my previous post Field Extension problem beyond $\mathbb C$. Notation: $M(\mathbb C):=$ Field of all meromorphic functions on $\mathbb C$, ...
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Given a normal basis $\{g(a)| g\in Gal(L/K)\}$ of $L/K$ finite Galois, can we express $g(a)g'(a)$ relative to this basis?

Let $L/K$ be a finite Galois extension. The normal basis theorem says that there is an element $a\in L$ such that $\{ g(a) | g\in \text{Gal}(L/K)\}$ is a basis of $L$ as a $K$-vector space. Let ...
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1answer
663 views

Need help understanding separable polynomials

I have the following definition of what it means for a polynomial to be separable: Let $E$ be a field and $P \in E[x]$ be irreducible. Then we say $P$ is separable if it has no repeated root in ...
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309 views

Field Isomorphisms

Suppose $F/L$, $F'/L$, $L/K$ finite extensions of fields. If $F$, $F'$ isomorphic over $K$ then does it follow that they are isomorphic over $L$? I think probably not, but I can't come up with a ...
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Galois Extensions generated by Algebraic Representations

$\newcommand{\Q}{\mathbb Q}$ Let $F=\mathbb Q^{ab} \subset \mathbb C$, i.e. the algebraic numbers. Let $G$ be a finite group of order $n$ and let $\phi: G \rightarrow GL_m(F)$ be a representation. ...
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Galois Group of $X^4+X^3+1$ over $\mathbb{Q}$

Just a quick sanity check. Am I right in thinking that the Galois group of $X^4+X^3+1$ over $\mathbb{Q}$ is isomorphic to $S_4$? It's irreducible (since it's irreducible mod 2) and so strictly ...
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Finding a minimal polynomial in char $2$.

[Some (useless) context: the following problem comes from a problem in Algebraic Geometry, where I have to show that a certain morphism $\textbf P^2\to \textbf A^2$ is inseparable of degree $2$.] Let ...
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1answer
258 views

General Primitive Element Theorem

I have a proof of the primitive element theorem for subfields $K$ of $\mathbb{C}$ which relies on there being infinitely many elements in $K$. In a book I've been reading it states that every finite ...
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1answer
263 views

Field Extension problem beyond $\mathbb C$

There are lots of fields between $\mathbb C$ and Meromorphic Functions on $\mathbb C$. For example set of "All Even Meromorphic Functions on $\mathbb C$'' is a subfield between $\mathbb C$ and ...
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Abel-Ruffini Theorem Clarification

Let $n \geq 5$. I want to show that $\exists p\in \mathbb{Q}[X]$ of degree $n$ with roots which are not possible to find via radicals and rational functions from the coefficients of $p$. I've got the ...
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1answer
235 views

Solubility of a Galois Group

going over some past papers with no answers and would like a bit of help if possible.. I've shown that for p a prime number then $x^p-1 \in K[x]$ is abelian where K is a subfield of $\mathbb{C}$. ...
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Find all finite fields k whose subfields form a chain: if $k'$ and $k''$ are subfields of $k$, then either $k' \subseteq k''$ or $k'' \subseteq k'$.

Find all finite fields $k$ whose subfields form a chain: that is, if $k'$ and $k''$ are subfields of $k$, then either $k' \subseteq k''$ or $k'' \subseteq k'$. So, I understand that I'm trying to ...
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188 views

Two Equivalent Notions of Algebraic Simple Extension (Proof)

When reading texts about field extensions I've come across the following two definitions for the simple extension $K(\alpha)$ where $\alpha$ some algebraic number over $K$. They are (1) $K(\alpha)$ ...
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Prove that $\mathrm{Aut}(L(\zeta_n)/K(\zeta_n))$ is a subgroup of $\mathrm{Aut}(L/K)$

Let $L/K$ be a Galois extension of fields and let $\zeta_n$ be a primitive $n$th root of unity in some field extension of $L$, where $n$ is not divisible by the characteristic. Prove that ...
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Proof that a polynomial is irreducible in $\mathbb{Q}$

Let $p$ be a prime number, and let $m,k_1,\ldots,k_{p-2}$ be even numbers. Define the polynomial $h(x)=(x^2+m)(x-k_1)\cdots(x-k_{p-2})$ and $r=\min \{|h(a)|\mid a\in\mathbb{R},h'(a)=0\}$. Under these ...
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Galois Group of $(x^3-5)(x^2-3)$

I am having some trouble calculating the Galois group (over $\mathbb{Q}$) of $(x^3-5)(x^2-3)$. I can see the splitting field is ...
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Show the norm map is surjective

Let $K/F$ be an extension of finite field. Show that the norm map $N_{K/F}$ is surjective. Here is what I have so far: Since $F$ is a finite field and $K/F$ is a finite extension of degree $n$, so ...
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Is the size of the Galois group always $n$ factorial?

I am studying field theory, and I just started the chapter on Galois theory. Since a Galois extension is the splitting field of some polynomial $p(x)$ and this polynomial have exactly $n$ roots in ...
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Calculating fixed field of a particular action

Let $K = \mathbb F_p(x)$, and let $H = \left\{\begin{pmatrix} d & a \\ 0 & 1 \end{pmatrix} \ \big| \ a \in \mathbb F_p, d \in \mathbb F_p^\times\right\}$ be a group under multiplication which ...
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Why is this extension of Galois?

Let $F$ be a subextension of $\mathbb{C}$ maximal with respect to not containing $\sqrt2$. Let $K/F$ be a finite extension with $K\subset\mathbb{C}$. Then $K/F$ is of Galois and $[K:F]$ is a power of ...
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Example of field $K$ with $\mathrm{char}(K) > 0 $, such that $[K(\alpha):K] = [K(\beta):K]$ but $K(\alpha) \not \cong K(\beta)$

I'd like to fine an example of field $K$ and elements $\alpha, \beta$ such that $\mathrm{char}(K) = p> 0 $, $[K(\alpha):K] = [K(\beta):K]$ but $K(\alpha) \not \cong K(\beta)$. This obviously can't ...
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44 views

Why $H\neq N_G(H)$?

Let $K$ be a field, $f(x)$ a separable irreducible polynomial in $K[x]$. Let $E$ be the splitting field of $f(x)$ over $K$. Let $\alpha,\beta$ be distinct roots of $f(x)$. Suppose ...
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846 views

What is a maximal abelian extension of a number field and what does its Galois group look like?

How does one know that a number field $K$ has a maximal abelian extension (unique up to isomorphism) $K^{\text{ab}}$? I've read proofs involving Zorn's lemma that it has an algebraic closure (And ...
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201 views

Order and generator of the Galois group of an extension of finite fields

I'm trying to find the order and describe a generator of the group $$\mathrm{Aut}_{\mathrm{GF}(2^3)}(\mathrm{GF}(2^{12}))$$ It's clear that the order is 4, but how would you describe the generator? ...
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91 views

why this extension doesn't contain a subextension of degree 2?

Consider the polynomial $f(x)=x^4+6x^3+32x^2+17x-15$ and let $\alpha\in\mathbb{C}$ be a root of $f$. How can I show that $\mathbb{Q}(\alpha)$ has no subfield of degree 2 over $\mathbb{Q}$? I have an ...
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107 views

Endomorphisms and Automorphisms

Let $L$ be a field extension of $K$. Consider the set $\operatorname{End}_KL$ of all functions from $L$ to $L$ which are linear over $K$. A subset of $\operatorname{End}_KL$ is $\Gamma(L:K)$, the ...
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Need help determining the Galois group of an extension

In an assignment I got I have been asked to try to determine when $E/F$ is a Galois Extension, and determine the Galois group of such an extension. $\textbf{Context:}$ $F$ is any field of ...